Book contents
- Frontmatter
- Contents
- Foreward
- Preface
- Chapter 1 Basic Ideas of Scientific Computing
- Chapter 2 Governing Equations in Fluid Mechanics
- Chapter 3 Classification of Quasi-Linear Partial Differential Equations
- Chapter 4 Waves and Space–Time Dependence in Computing
- Chapter 5 Spatial and Temporal Discretizations of Partial Differential Equations
- Chapter 6 Solution Methods for Parabolic Partial Differential Equations
- Chapter 7 Solution Methods for Elliptic Partial Differential Equations
- Chapter 8 Solution of Hyperbolic PDEs: Signal and Error Propagation
- Chapter 9 Curvilinear Coordinate and Grid Generation
- Chapter 10 Spectral Analysis of Numerical Schemes and Aliasing Error
- Chapter 11 Higher Accuracy Methods
- Chapter 12 Introduction to Finite Volume and Finite Element Methods
- Chapter 13 Solution of Navier–Stokes Equation
- Chapter 14 Recent Developments in Discrete Finite Difference Computing
- Exercises
- References
- Index
Chapter 6 - Solution Methods for Parabolic Partial Differential Equations
Published online by Cambridge University Press: 05 January 2014
- Frontmatter
- Contents
- Foreward
- Preface
- Chapter 1 Basic Ideas of Scientific Computing
- Chapter 2 Governing Equations in Fluid Mechanics
- Chapter 3 Classification of Quasi-Linear Partial Differential Equations
- Chapter 4 Waves and Space–Time Dependence in Computing
- Chapter 5 Spatial and Temporal Discretizations of Partial Differential Equations
- Chapter 6 Solution Methods for Parabolic Partial Differential Equations
- Chapter 7 Solution Methods for Elliptic Partial Differential Equations
- Chapter 8 Solution of Hyperbolic PDEs: Signal and Error Propagation
- Chapter 9 Curvilinear Coordinate and Grid Generation
- Chapter 10 Spectral Analysis of Numerical Schemes and Aliasing Error
- Chapter 11 Higher Accuracy Methods
- Chapter 12 Introduction to Finite Volume and Finite Element Methods
- Chapter 13 Solution of Navier–Stokes Equation
- Chapter 14 Recent Developments in Discrete Finite Difference Computing
- Exercises
- References
- Index
Summary
Introduction
Initial developments in computing were dominated by two classes of problems: (i) the jury or the boundary value problems – typically classified as elliptic partial differential equation in Chapter 3; (ii) the evolution or the initial–boundary value problems which are represented by parabolic and hyperbolic partial differential equations. In fact, the solution methods for heat equation (a parabolic partial differential equation) were central to the early development of the subject. These classical approaches are discussed in this chapter, with additional insight brought through spectral analysis of the schemes. It is noted that the stability analysis of numerical schemes was developed with respect to heat equation by von Neumann, as described in [41, 53]. This was considered a major milestone in the development of the subject. But, the readers' attention is also drawn to the correct analysis advanced recently, as described in [259] and Chapter 8, with respect to 1D convection equation.
In fluid dynamics, a major milestone was the introduction of boundary layer concept by Ludwig Prandtl in 1904, which dominated fluid dynamics studies. Readers are referred to [209] for details of the development. Boundary layer equation is an example of parabolic partial differential equation.
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- High Accuracy Computing MethodsFluid Flows and Wave Phenomena, pp. 92 - 105Publisher: Cambridge University PressPrint publication year: 2013